By: Christian Mayrhofer, Manager R&D—Advanced Machine & Engineering/AMSAW

It is no easy task to move material stock that spans over 12 meters in length while weighing a few tons into a sawing system. For this process, it is common to use forklifts or gantry cranes to move the material onto a loading table or transfer mechanism. The danger any plant manager faces, is the potential that an inexperienced crane or lift truck operator may accidentally drop heavy materials from a higher distance, damaging the material load table. This uncontrollable interaction may also put the rest of the material handling system at risk for severe abuse. A poka-yoke system is needed to guarantee high Overall Operations Effectiveness (OOE).

**Figure 1** An example of severe material load table damage experienced from a heavy material drop.

## Why Material Load Tables are Over-Engineered

Material load tables and transfer mechanisms are often over-engineered with excessive material in an attempt to avoid severe damage from dropped loads. Although the strength of the load table may be improved, this may increase the cost—a cost that gets passed onto the customer. Another approach to avoid increased costs when designing material handling systems is to use advanced engineering practices.

Modern CAD technology, such as SolidWorks, provides powerful tools to calculate and analyze mechanical structures. These programs make it easier to add and test materials that fortify structures in CAD software. The advantage of working in the virtual world is that you can test a variety of materials without adding more cost to the prototyping process.

Within these CAD programs, features such as Finite Element Analysis (FEA), can simulate the effects of shock load before the prototype product is even produced. This allows for early modifications in the design process at little-to-no cost, while ensuring the best strength-to-weight ratios at the lowest cost for the customer.

## Theory and Analytical Example

By using the simplifying assumptions of strain energy and the principle of conservation of energy, one can presume that the potential energy before the impact, the kinetic energy right at the impact, and the stored elastic energy in a structure (spring) are equal. In the following formula, \(\delta_{max}\) is the maximum displacement of the spring; *c* is the spring rate; *m* is the falling mass;* g* is the acceleration of gravity; and *h* is the starting height:

$$mg (h+\delta_{max}) = \left(\frac{c*\delta^2_{max}}{2}\right)$$

The formula for introducing displacement under a static loading condition \(\delta_{st}\) looks like:

$$\delta_{max} = \delta_{st}\Bigl[1 + \sqrt {(1 + \frac{2h}{\delta_{st}})}\Bigr]$$

The same formula can be written in a different form when applying Hooke’s Law — \(F_{max}\) is the maximum contact force and \(F_{st}\) is the static force:

$$F_{max} = F_{st}\Bigl[1 + \sqrt {(1 + \frac{2h}{\delta_{st}})}\Bigr]$$

This confirms that a quasi-static calculation with an amplification factor (impulse factor) is possible, and conducting dynamic simulations as a first approach is not necessary. This formula can also be applied to a static displacement calculation of a beam with the use of the Euler-Bernoulli beam theory.

**Example:**

Let us assume a 100 kg mass is hitting a support beam and the contact force and resulting stress needs to be known: *E* (Young’s modulus) is 210 GPa; *h* is 25 mm;* I* is 1 m; and the beam cross section is a 50 mm x 25 mm rectangle.

**Figure 2** A visual representation calculating the contact force and resulting stress of a dropped support beam.

Moment of inertia for a rectangular profile is:

$$I = \frac{bh^3}{12} = 65*10^{-9}m^4$$

The static displacement is:

$$\delta_{st} = \frac{mgl^3}{48EI} = 65*10^{-9}m$$

Bending stress due to static load—*M* is the maximum bending moment; *W* is the moment of resistance:

$$\sigma_{st} = \frac{M}{W} = \frac{F\frac{L}{2}}{\frac{bh^2}{6}} = 94 MPa$$

The impact factor is:

$$n = 1 + \sqrt {(1 + \frac{2h}{\delta_{st}})} = 6.7$$

This demonstrates that the impact force from a 25 mm height will be seven times higher while the static load of the mass is 981 N. The bending stress will rise in the same fashion.

The formula for the impact factor shows that an increase in stiffness reduces the static displacement. This increases both the impact factor and impact force. The result shows the paradigm that is generally true from machine tools: the stiffer the better; however, this is not necessarily true when dealing with shock loads.

## Verification of Durability

The Transient Nonlinear Dynamic Analysis in FEA software verifies the product in an early stage and conducts damage analysis of it. This provides insight on the contact, reaction forces, and the stress as a function of time. The impact takes place in a matter of milliseconds, making the adequate discretization of space and time crucial. The boundary condition represents the interface to the floor. The initial condition is the impact velocity \((v = \sqrt {2gh})\)—with *g* being the gravity constant and *h* the starting height from where a steel billet is dropped.

**Figure 3 & 4** FEA software analysis of heavy load drop contact, reaction forces, and stress as a function of time.

## Improved Ways to Design Material Load Tables

With today’s technology, there are improved ways to design material load tables so they are not overloaded with costs. Generally, designers have two choices:

- Over-engineer the structure to withstand any heavy shock load, increasing the cost of the load table for the customer.
- Through the use of FEA software, design the structure to allow elastic flexing while avoiding the effects of plastic deformation.

We have all have dropped our smartphones. To reduce the risk of cracking or shattering the screen, we have placed the phone within a shock absorbing case. This simple approach reduces the stiffness and impact forces that correspond with the stresses of the phone when dropped. The same is true for dropping material onto a material load table.

*It's only fair to share...*

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